11/26/17. directed graphs. CS 220: Discrete Structures and their Applications. graphs zybooks chapter 10. undirected graphs

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1 directed graphs S 220: iscrete Structures and their pplications collection of vertices and directed edges graphs zybooks chapter 10 collection of vertices and edges undirected graphs What can this represent? undirected vs directed edges Note the differences in how directed and undirected edges are represented: 1

2 directed / undirected graphs in applications constraint graphs irected or undirected graph? ü ü ü ü Facebook friend graph The "follow" graph The "like" graph Knowledge graph onsider the problem of classroom scheduling: given a set of classes and their times, assign them to classrooms without conflicts. xample: lass : MWF, 3:00PM - 4:00PM lass : W, 2:00PM - 4:00PM lass : F, 3:30PM - 5:00PM lass : MWF, 2:30-3:30PM Which is the constraint graph for this scheduling problem? undirected graphs terminology xamples: What is the set of edges for the graph on the right? G=(V, ) Vertices dges dges Two vertices are adjacent if they are connected by an edge. The vertices are the endpoints of an edge n edge is incident on two vertices. Vertices/ Nodes u e v Two vertices are neighbors if they are connected by an edge The number of neighbors of a vertex is its degree. 2

3 question undirected graphs What is the degree of c? b c d No "parallel" edges No self loops self loop: an edge that connects a vertex to itself a f e g simple graph: no self loops and no two edges that connect the same vertices. We will focus on simple graphs. the handshake theorem Theorem: Let G=(V,) be an undirected graph. Then deg(v) = 2 v V subgraphs graph H = (V H, H ) is a subgraph of a graph G = (V G, G ) if V H V G and H G. 3

4 complete graphs complete graphs is a simple graph that has an edge between every pair of vertices. cycles The cycle n, n 3, consists of n vertices v 1, v 2,, v n and n edges {v 1, v 2 }, {v 2, v 3 },, {v n-1, v n }, {v n, v 1 }. The complete graph with n vertices is denoted by K n K 4 : omplete Graph the n-dimensional hypercube regular graphs regular is a graph in which all vertices have the same degree. The Hypercube Q

5 looks can be misleading adjacency matrix of a graph onsider the following two graphs: re they the same? mapping of vertex labels to array indices Label Index For undirected graph, what would the adjacency matrix look like? djacency matrix: n x n matrix with entries that indicate if an edge between two vertices is present question adjacency matrix Is this the adjacency matrix of an undirected graph?. Yes. No djacency Matrix djacency matrix for an undirected graph

6 question djacency Matrix adjacency list for a directed graph oes this graph have self loops?. Yes. No Index Label adjacency list for an undirected graph which implementation Index Label mapping of vertex labels to list of edges djacency matrix dges are entries in a square matrix size: V 2 values: 1/0 to indicate presence/absence of edge in (un)directed graph useful for dense graphs djacency list linked-list of out-going edges per vertex useful for sparse graphs 6

7 which implementation walks Which implementation best supports common graph operations: Is there an edge between vertex i and vertex j? Find all vertices adjacent to vertex j What's the big O for each of these operations? v 0 e 1 walk from v 0 to v l in an undirected graph G is a sequence of alternating vertices and edges that starts and ends with a vertex: v 0,{v 0,v 1 },v 1,{v 1,v 2 },v 2,...,v l 1,{v l 1,v l },v l Which best uses space? v 1 e 2 v 2 e 3 v 3 walk can also be denoted by the sequence of vertices: v 0,v 1,...,v l. The sequence of vertices is a walk only if {v i-1, v i } for i = 1, 2,...,l. The length of a walk is l, the number of edges in the walk. walks, circuits, paths, cycles walks, circuits, paths, cycles v 0 e 1 v 1 e 2 v 3 circuit is a walk in which the first vertex is the same as the last vertex. sequence of one vertex, denoted <a>, is a circuit of length 0. walk is a path if no vertex is repeated in the walk. circuit is a cycle if there are no other repeated vertices, except the first and the last. circuit is a walk in which the first vertex is the same as the last vertex. walk is a path if no vertex is repeated in the walk. circuit is a cycle if there are no other repeated vertices, except the first and the last. ² What is the length of the longest possible walk in a graph with n vertices? ² What is the length of the longest possible path in a graph with n vertices? e 3 Same as in directed graphs. ² What is the length of the longest possible circuit in a graph with n vertices? v 2 ² What is the length of the longest possible cycle in a graph with n vertices? 7

8 connected components n undirected graph is called connected if there is a path between every pair of vertices. connected component is a maximal set of vertices that is connected. The word "maximal" means that if any vertex is added to a connected component, then the set of vertices will no longer be connected. example How many connected components does this graph have? a e c d f b The facebook graph u 721 million active accounts u 68.7 billion friendship edges (median number of friends = 99) u The largest connected component of facebook users contains 99.9% of the users u verage distance between any pair of users: 4.7 source: 8